Heat capacity of a solid

《Heat capacity of a solid》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、TheHeatCapacityofaSolid6.1INTRODUCTIONTheinvestigationoftheheatcapacityofsolidsisimportantinthestudyofcondensedmatter.Thisexample,likethediatomiccase,illustratestheshortcomingsofclassicalkinetictheoryandtheneedforstatisticalmechanicsandquantumtheorytoprovideanswersthatagreewithexperimentalre

2、sults.In1819,DulongandPetitobservedthatthespecificheatcapacityat4#1#1constantvolumeofallelementarysolidsisapproximately2.49"10JkilomoleKattemperaturesnearroomtemperature.Thisresultcanbeexplainedbytheprincipleofequipartitionofenergybytreatingeveryatomofthesolidasalinearoscillatorwithsixdegree

3、soffreedomandthenassociatinganenergyof1kTwitheachdegreeoffreedom.Then2c=6(R2)=3R,inagreementwiththeobservationofDulongandPetit.vHowever,additionalmeasurementsshowedthatthespecificheatofsolidsvarieswithtemperature,decreasingtozeroasthetemperatureapproacheszero.Thisbehaviorcannotbeexplainedbyt

4、he“freezing”ofdegreesoffreedomwhenthetemperatureisdecreasedsincethespecificheatvariesgraduallywithtemperatureanddoesnotexhibitabruptjumpsbyany1multipleofR(incontrasttothespecificheatofadiatomicgas).Evenatroomtemperature2thespecificheatcapacitiesofcertainsubstancessuchasberyllium,boron,carbon

5、,andsiliconwerefoundtobemuchsmallerthan3R.Quantumstatisticsisneededtoexplainthesediscoveries.6.2EINSTEIN’STHEORYOFTHEHEATCAPACITYOFASOLIDItwasEinsteinwhodevelopedthefirstreasonablysatisfactorytheoryofthespecificheatcapacityofasolid.HeassumedthatthecrystallatticestructureofasolidcomprisingNat

6、omscanbetreatedasanassemblyof3Ndistinguishableone-dimensionaloscillators(threeoscillatorsforeachatomsincetheatomsofasolidarefreetomoveinthreedimensions).FromEquation(5.8),theinternalenergyofasolidmadeupofNatomsis1$11'U=3Nk"E&+"T),(6.1)%2eE#1(where"istheEinsteintemperaturegivenby"#h$k.Theheat

7、capacityatconstantEEvolumeis2)T#"U&#)&eEECV=%(=3Nk%(2(6.2)$"T'V$T'(e)ET*1)FortemperaturesverylargecomparedwiththeEinsteintemperature,C"3Nk=3nR.(6.3)VThusthehightemperaturelimitofEinstein’sequationgivesthevalueofDulongandPetit.Thefailureoftheirlawbe